Noether's Theorem: Symmetry to Conservation

What you are seeing: a particle in an attractive central potential $V = -1/r$ . Rotation symmetry of the Lagrangian implies $L_z$ is conserved. Adding a small angular term $\varepsilon \cos(2\theta)/r$ breaks the symmetry and you can watch $L_z$ start to drift.

Figure 1. Orbit (left) and conserved-quantity trace (right).

ε (sym-break)0.08

v_init1.00

WHAT TO TRY

Start from the broken default: the cos(2 theta) term tilts the potential, the orbit precesses into a rosette, and the L_z(t) trace visibly oscillates. Rotation symmetry is gone, so its conserved quantity is gone.
Drag epsilon down to zero: the potential becomes rotationally symmetric again, the orbit closes, and L_z flattens into a straight line. Symmetry implies a conserved quantity, that is Noether theorem.
Compare the L_z and energy traces: with the symmetry intact both are flat, and the moment you tilt the potential the conserved quantity tied to that symmetry is the one that moves.